Sharp Convergence for Degenerate Langevin Dynamics

The paper proves a sharp convergence profile for the 1D small-noise Langevin SDE dX^ε_t = −V'(X^ε_t) dt + √ε dB_t under convexity, an even potential with a degenerate fixed point at 0, a precise local expansion V′(λz) ~ C0λ^{1+α}|z|^{1+α}sgn z, and polynomial growth at infinity. It shows that, after the correct space–time rescaling (b_ε, a_ε) given by solving √(ε a_ε)/b_ε = 1 and a_ε b_ε^α = 1, one has b_ε = ε^{1/(2+α)} and a_ε = ε^{-α/(2+α)} (derived in the multi-scale analysis and used throughout the proofs), and the total variation profile converges to d_TV(Y_t(sgn(x)∞), Y_∞) where Y solves dY_t = −C0|Y_t|^{1+α}sgn(Y_t)dt + dW_t with entrance from ±∞. The argument proceeds via a decoupling inequality and two limit replacements: convergence of the rescaled finite-time marginals and of the rescaled invariant laws (both in total variation). See the multi-scale derivation and its consequences in (2.6)–(2.8) and the use of ε a_ε = ε^{2/(2+α)} in the proofs, as well as the decoupling inequality (2.15), Proposition 2.1, and Lemma 3.3 on invariant measures, culminating in Theorem 1.1 and its corollaries about no cutoff and mixing times . The candidate solution uses the wrong time scale a_ε = ε^{-2/(2+α)} and then informally “divides by A_ε” in the Fokker–Planck equation, which effectively reintroduces the correct scale but contradicts the stated choice and the claimed limit for d_TV(X^ε_{t a_ε}(x), μ^ε). It also relies on an unproven TP2/monotone likelihood-ratio argument and asserts dominated/L^1 local convergence without establishing the uniform bounds the paper derives via coupling and appendices. Hence, the model’s proof outline is incorrect on the central scaling and unsupported on key steps, while the paper’s proof is logically coherent; the only caveat is a likely typo in Theorem 1.1 where (1.2) prints a_ε = ε^{-2/(2+α)} although the analysis and proofs consistently use a_ε = ε^{-α/(2+α)} (compare (1.2) with (2.7)–(2.8) and subsequent uses of ε a_ε) .